Nuclear Gravitational Spectroscopy

• Nuclear gravitational spectroscopy is a suite of techniques that measure gravity-induced frequency shifts in both ultracold neutron bound states and nuclear transitions.
• It employs mechanical oscillations, magnetic-field gradients, and Ramsey-type interferometry to probe energy differences within the peV range and beyond.
• The methods enable precision tests of Newtonian gravity and general relativity, constraining new physics such as chameleon fields and informing astrophysical models.

Searching arXiv for the cited works to ground the article in the provided literature. arxiv_search.query({"search_query":"id:(Sedmik et al., 2019) OR id:(Pignol et al., 2014) OR id:(Baeßler et al., 2015) OR id:(Abele et al., 2014) OR id:(Kulin et al., 2015) OR id:(Röhlsberger, 18 Apr 2026)","start":0,"max_results":10}) arxiv_search.search({"query":"(Sedmik et al., 2019, Pignol et al., 2014, Baeßler et al., 2015, Abele et al., 2014, Kulin et al., 2015, Röhlsberger, 18 Apr 2026)","max_results":10}) Nuclear gravitational spectroscopy denotes a family of spectroscopic methods in which gravity enters the measured frequency or energy splitting, either by quantizing a neutron’s external center-of-mass motion in the Earth’s field or by gravitationally shifting an internal nuclear transition frequency. In the neutron-based literature, the dominant meaning is gravity resonance spectroscopy (GRS) with ultracold neutrons above mirrors, where the spectroscopic object is the neutron’s gravitationally bound motional state rather than an internal nuclear excitation (Sedmik et al., 2019). In a narrower and more literal nuclear sense, the term now also includes proposed measurements of the gravitational redshift of a nuclear resonance, exemplified by heterodyne interferometry of the $^{57}$Fe Mössbauer transition (Röhlsberger, 18 Apr 2026).

1. Terminological scope and historical framing

The expression has two technically distinct usages. In the qBounce and GRANIT programs, the neutron is a neutral, massive quantum probe whose vertical motion becomes quantized above a horizontal mirror in the Earth’s linear potential $V(z)=m_ngz$. Spectroscopy then addresses transitions between gravitationally bound quantum states with energies in the peV range and transition frequencies in the acoustic/hundreds-of-hertz range; these are external motional states, not standard nuclear excitations (Sedmik et al., 2019). By contrast, a newer usage concerns tests of the gravitational redshift of an actual nuclear transition frequency, where two identical nuclear absorbers at different heights acquire slightly different resonance frequencies according to $\delta\omega_{\rm GR}=\omega_0 gh/c^2$ (Röhlsberger, 18 Apr 2026).

This distinction is important because much of the neutron literature is explicitly not spectroscopy of internal nuclear levels. The papers on qBounce, GRANIT, and related ultracold-neutron platforms repeatedly define the subject as spectroscopy of the neutron’s center-of-mass motion in an external gravitational potential (Abele et al., 2014). A plausible implication is that “nuclear gravitational spectroscopy” is best treated as an umbrella term whose subfields differ in whether gravity acts on an external motional degree of freedom or on an internal nuclear transition.

Historically, the nuclear-transition branch connects back to Mössbauer redshift measurements, while the neutron branch emerged from observation of gravitationally bound ultracold-neutron states and evolved into resonance spectroscopy. The newer heterodyne proposal is presented as a way to revive nuclear-sector redshift tests by converting the problem from energy-domain detection to time-domain interferometry (Röhlsberger, 18 Apr 2026).

2. Quantum-bouncer foundation of neutron gravitational spectroscopy

The basic neutron system is the “quantum bouncing neutron”: an ultracold neutron above a reflecting mirror, with a hard-wall boundary at $z=0$ and a linear gravitational potential upward. The stationary Schrödinger equation is written as

$$\left[-\frac{\hbar^2}{2m_n}\frac{d^2}{dz^2}+m_ngz\right]\psi(z)=E\psi(z),\qquad \psi(0)=0,$$

and its solutions are Airy-function eigenstates (Sedmik et al., 2019). Equivalent notation in the GRANIT treatment writes the Hamiltonian as

$$\hat H_0=\frac{\hat p_z^2}{2m}+mg\hat z+V_{\rm mirror}(z),$$

with $V_{\rm mirror}(z)=\infty$ for $z\le 0$ and $0$ for $z>0$ (Pignol et al., 2014).

The characteristic gravitational length scale is

$V(z)=m_ngz$0

and one paper gives $V(z)=m_ngz$1, while the corresponding characteristic frequency scale is

$V(z)=m_ngz$2

(Pignol et al., 2014). In the qBounce notation, one may equivalently write

$V(z)=m_ngz$3

and the low-lying levels are non-equidistant, with one reported sequence

$V(z)=m_ngz$4

(Sedmik et al., 2019).

Because the levels are non-equidistant, each transition has a unique frequency,

$V(z)=m_ngz$5

Representative low-lying frequencies discussed experimentally include $V(z)=m_ngz$6, $V(z)=m_ngz$7, and, in the Ramsey proof-of-principle, $V(z)=m_ngz$8 for the $V(z)=m_ngz$9 transition (Pignol et al., 2014).

A central motivation for using neutrons is their unusually clean systematics profile. The neutron has negligible electric charge, a tiny electric dipole moment, and very small polarizability, so it is much less affected than atoms or macroscopic bodies by electrostatic, van der Waals, and related electromagnetic backgrounds (Sedmik et al., 2019). The overview article contrasts this explicitly with atom-surface experiments by quoting, for $\delta\omega_{\rm GR}=\omega_0 gh/c^2$0 at $\delta\omega_{\rm GR}=\omega_0 gh/c^2$1, a Casimir potential $\delta\omega_{\rm GR}=\omega_0 gh/c^2$2, which is already on the scale of neutron gravitational energies (Abele et al., 2014).

3. Resonant driving architectures and spectrometer geometries

In neutron gravitational spectroscopy, transitions are induced by a periodic perturbation with nonzero matrix elements between Airy states. Standard qBounce GRS uses mechanical oscillation of a mirror, so that the moving boundary condition drives transitions. The time-dependent Hamiltonian is written as

$\delta\omega_{\rm GR}=\omega_0 gh/c^2$3

and, because the mirror Fermi potential is large, the mirror can be treated as an effectively infinite wall with moving boundary

$\delta\omega_{\rm GR}=\omega_0 gh/c^2$4

(Sedmik et al., 2019). Expanding the state as $\delta\omega_{\rm GR}=\omega_0 gh/c^2$5, the amplitudes obey

$\delta\omega_{\rm GR}=\omega_0 gh/c^2$6

A complementary GRANIT approach uses an oscillating magnetic-field gradient rather than mechanical vibration. Under the adiabatic spin-transport approximation, the neutron feels an effective scalar potential

$\delta\omega_{\rm GR}=\omega_0 gh/c^2$7

and for a nearly linear, oscillatory field magnitude $\delta\omega_{\rm GR}=\omega_0 gh/c^2$8, the perturbation becomes

$\delta\omega_{\rm GR}=\omega_0 gh/c^2$9

with Rabi angular frequency

$z=0$0

(Pignol et al., 2014). The relevant matrix elements are

$z=0$1

The GRANIT wire-array design supports two operating modes with the same hardware. In DC mode, the magnetic gradient is static in time but periodic in space with period $z=0$2, so a neutron with horizontal velocity $z=0$3 experiences an excitation frequency $z=0$4. In AC mode, the gradient is approximately homogeneous in space and oscillating in time, so the resonance is obtained by directly scanning the drive frequency (Pignol et al., 2014). The AC mode is notable because the excitation frequency is twice the current-driving frequency owing to a $z=0$5 dependence in the field magnitude, with Fourier coefficients $z=0$6 and $z=0$7 quoted for one benchmark configuration (Pignol et al., 2014).

The geometries differ correspondingly. GRANIT is a flow-through spectrometer in which neutrons traverse state preparation, transition, analyzer, and detector stages. Its transition region has length $z=0$8, giving a mean interaction time $z=0$9 for the average velocity $$\left[-\frac{\hbar^2}{2m_n}\frac{d^2}{dz^2}+m_ngz\right]\psi(z)=E\psi(z),\qquad \psi(0)=0,$$0, and the analyzer slit is about $$\left[-\frac{\hbar^2}{2m_n}\frac{d^2}{dz^2}+m_ngz\right]\psi(z)=E\psi(z),\qquad \psi(0)=0,$$1 high (Pignol et al., 2014). qBounce’s Ramsey implementation instead uses a dedicated five-region spectrometer at PF2, with regions 2 and 4 as oscillating interaction zones, state selectors in regions 1 and 5 at slit height $$\left[-\frac{\hbar^2}{2m_n}\frac{d^2}{dz^2}+m_ngz\right]\psi(z)=E\psi(z),\qquad \psi(0)=0,$$2, and a measured horizontal velocity spectrum centered around about $$\left[-\frac{\hbar^2}{2m_n}\frac{d^2}{dz^2}+m_ngz\right]\psi(z)=E\psi(z),\qquad \psi(0)=0,$$3 (Sedmik et al., 2019).

A distinct but related gravitational spectrometer compares the free-fall energy gain of ultracold neutrons to a discrete energy quantum $$\left[-\frac{\hbar^2}{2m_n}\frac{d^2}{dz^2}+m_ngz\right]\psi(z)=E\psi(z),\qquad \psi(0)=0,$$4 generated by a moving diffraction grating. In that device the neutron energy spectrum is split into sidebands

$$\left[-\frac{\hbar^2}{2m_n}\frac{d^2}{dz^2}+m_ngz\right]\psi(z)=E\psi(z),\qquad \psi(0)=0,$$5

and the experiment compares $$\left[-\frac{\hbar^2}{2m_n}\frac{d^2}{dz^2}+m_ngz\right]\psi(z)=E\psi(z),\qquad \psi(0)=0,$$6 against $$\left[-\frac{\hbar^2}{2m_n}\frac{d^2}{dz^2}+m_ngz\right]\psi(z)=E\psi(z),\qquad \psi(0)=0,$$7 to test the weak equivalence principle for neutrons (Kulin et al., 2015). This suggests that neutron gravitational spectroscopy also includes free-fall spectrometry, not only bound-state spectroscopy.

4. Ramsey-type coherence methods and interferometric extensions

A major methodological advance in the neutron branch is the transfer of Ramsey’s separated oscillatory fields method to gravitationally bound neutron states. In the qBounce implementation, a state selector prepares mainly low gravitational states, the first oscillating region applies a $$\left[-\frac{\hbar^2}{2m_n}\frac{d^2}{dz^2}+m_ngz\right]\psi(z)=E\psi(z),\qquad \psi(0)=0,$$8 pulse on a chosen transition, a passive mirror region allows free evolution, and a second oscillating region applies a second $$\left[-\frac{\hbar^2}{2m_n}\frac{d^2}{dz^2}+m_ngz\right]\psi(z)=E\psi(z),\qquad \psi(0)=0,$$9 pulse with adjustable relative phase $$\hat H_0=\frac{\hat p_z^2}{2m}+mg\hat z+V_{\rm mirror}(z),$$0 (Sedmik et al., 2019). In the paper’s two-state notation for the $$\hat H_0=\frac{\hat p_z^2}{2m}+mg\hat z+V_{\rm mirror}(z),$$1 transition,

$$\hat H_0=\frac{\hat p_z^2}{2m}+mg\hat z+V_{\rm mirror}(z),$$2

with $$\hat H_0=\frac{\hat p_z^2}{2m}+mg\hat z+V_{\rm mirror}(z),$$3 in the free-evolution region.

The key Ramsey signature is phase coherence, not merely state transfer. If coherence were lost, the final transmission would be independent of the relative phase between the oscillating regions. If coherence is preserved, the controlled phase offset $$\hat H_0=\frac{\hat p_z^2}{2m}+mg\hat z+V_{\rm mirror}(z),$$4 modulates the final transmission: for $$\hat H_0=\frac{\hat p_z^2}{2m}+mg\hat z+V_{\rm mirror}(z),$$5, the second $$\hat H_0=\frac{\hat p_z^2}{2m}+mg\hat z+V_{\rm mirror}(z),$$6 pulse undoes the first and revives the initial state, while for $$\hat H_0=\frac{\hat p_z^2}{2m}+mg\hat z+V_{\rm mirror}(z),$$7, the two pulses add coherently and complete transfer to $$\hat H_0=\frac{\hat p_z^2}{2m}+mg\hat z+V_{\rm mirror}(z),$$8, minimizing transmission (Sedmik et al., 2019). In proof-of-principle data taken at $$\hat H_0=\frac{\hat p_z^2}{2m}+mg\hat z+V_{\rm mirror}(z),$$9 with nominal oscillation velocity amplitude $V_{\rm mirror}(z)=\infty$0, a constant-transmission hypothesis was excluded at $V_{\rm mirror}(z)=\infty$1 confidence, establishing coherent Ramsey-type spectroscopy of gravitationally bound neutron states (Sedmik et al., 2019).

The practical advantage of the five-region Ramsey geometry is that it increases the total interaction time by roughly a factor of four compared with the earlier three-region setup and reduces sensitivity to broad horizontal velocity distributions near the central fringe (Sedmik et al., 2019). A plausible implication is that Ramsey geometry is especially well matched to ultracold-neutron beams, where flux is limited and time-of-flight spread is unavoidable.

A conceptually analogous development now exists for genuine nuclear-transition gravitational spectroscopy. The heterodyne proposal for $V_{\rm mirror}(z)=\infty$2Fe does not detect a gravitational redshift by scanning a resonance in the energy domain. Instead it uses a reference absorber with controlled Doppler detuning $V_{\rm mirror}(z)=\infty$3 to generate delayed beat signals

$V_{\rm mirror}(z)=\infty$4

$V_{\rm mirror}(z)=\infty$5

for lower and upper absorbers separated vertically by height $V_{\rm mirror}(z)=\infty$6 (Röhlsberger, 18 Apr 2026). The gravitational redshift is encoded as a slowly accumulating phase drift

$V_{\rm mirror}(z)=\infty$7

and the leading-order arm difference is

$V_{\rm mirror}(z)=\infty$8

This converts nuclear gravitational spectroscopy from energy-domain slope detection to time-domain interferometry. The proposal explicitly treats the method as a nuclear analogue of phase-sensitive interferometric spectroscopy familiar from Ramsey interrogation, while remaining distinct from optical recombination interferometry because the two arms are not recombined optically; the readout is based on subtraction of delayed intensities rather than path recombination (Röhlsberger, 18 Apr 2026).

5. Precision limits, systematic shifts, and new-physics sensitivity

As neutron gravitational spectroscopy matured, systematic frequency shifts became a central topic. A dedicated analysis identifies three principal classes: the Stern–Gerlach shift, the interference shift, and the spectator state shift (Baeßler et al., 2015). For magnetically driven transitions, the field expansion contains a static linear-gradient term $V_{\rm mirror}(z)=\infty$9, which modifies the effective gravitational acceleration to

$z\le 0$0

Because the quantum-bouncer energies scale as $z\le 0$1, the transition frequencies become spin dependent: $z\le 0$2 For the $z\le 0$3 example treated in that paper, the resulting shift is numerically large, about $z\le 0$4 for spin-up and $z\le 0$5 for spin-down (Baeßler et al., 2015).

The spectator state shift is more general. It arises because the periodic drive couples not only the resonant pair but also off-resonant “spectator” states, in close analogy with the AC Stark shift in atomic physics. Using Floquet theory, the paper derives a corrected resonance detuning

$z\le 0$6

and gives explicit second-order sums over all off-resonant couplings (Baeßler et al., 2015). For weak driving, representative shifts are modest: about $z\le 0$7–$z\le 0$8 Hz for $z\le 0$9, about $0$0 to $0$1 Hz for $0$2, and about $0$3–$0$4 Hz for $0$5; for mechanically induced $0$6, the shift is about $0$7 to $0$8 Hz (Baeßler et al., 2015). The paper also warns that in shorter, more strongly driven devices, the spectator-state shift can become roughly a 2% correction for the $0$9 resonance.

The interference shift reflects imperfect state preparation rather than true level dressing. In flow-through schemes, a preparation step may create a coherent superposition rather than a pure initial state, so the observed resonance line shape becomes asymmetric. For the nominal step height $z>0$0, one paper finds $z>0$1, sufficiently large to bias the apparent peak position if not modeled (Baeßler et al., 2015).

These systematic analyses are directly tied to the physics reach of the method. The overview article states that GRS allows tests of Newton’s gravity law at short distances and provides constraints on any possible gravity-like interaction (Abele et al., 2014). In that paper, the experiment is most sensitive to chameleon models in the range $z>0$2, excluding

$z>0$3

and improving the upper bound from precision atomic spectroscopy by five orders of magnitude (Abele et al., 2014). The qBounce Ramsey paper further frames improved level spectroscopy as a route toward stronger constraints on chameleon, symmetron, and axion interactions, while explicitly emphasizing that additional short-range interactions would shift the transition frequencies through

$z>0$4

(Sedmik et al., 2019).

The same precision logic appears in the free-fall spectrometer for equivalence-principle tests. There, the experimentally relevant relation is $z>0$5, extracted from phase-matched time-of-flight measurements, and the paper reports a collection rate allowing statistical uncertainty in the equivalence factor of order $z>0$6 per day after apparatus improvements (Kulin et al., 2015). Concrete systematic issues include admixture of the zero diffraction order into the selected $z>0$7 order, finite rotor-thickness timing correction, and possible oscillating background from inelastically scattered faster neutrons (Kulin et al., 2015).

6. Genuine nuclear-transition gravitational spectroscopy and broader related meanings

The most literal realization of nuclear gravitational spectroscopy is the proposed nuclear heterodyne interferometry of the $z>0$8Fe Mössbauer resonance. The experiment places two identical absorbers at different gravitational potentials and estimates a deviation parameter $z>0$9 defined by

$V(z)=m_ngz$00

In this framework the delayed waveform difference takes the template form

$V(z)=m_ngz$01

and the uncertainty is analyzed with Fisher information under Poisson counting statistics (Röhlsberger, 18 Apr 2026).

The benchmark $V(z)=m_ngz$02Fe configuration assumes $V(z)=m_ngz$03 per arm, heterodyne contrast $V(z)=m_ngz$04, effective linewidth $V(z)=m_ngz$05, effective phase accumulation time $V(z)=m_ngz$06, and finite-window factor $V(z)=m_ngz$07 for the observation window $V(z)=m_ngz$08 ns (Röhlsberger, 18 Apr 2026). Under these assumptions, the paper states that the $V(z)=m_ngz$09Fe gravitational redshift can be observed at $V(z)=m_ngz$10 in about 2 hours on a $V(z)=m_ngz$11 baseline, while an $V(z)=m_ngz$12 baseline gives $V(z)=m_ngz$13 in about 8 days, corresponding to percent-level precision on deviations from general relativity (Röhlsberger, 18 Apr 2026). This marks a qualitative shift from historical Mössbauer redshift measurements: the observable is no longer a compensated energy shift but a time-domain phase drift.

Beyond these laboratory platforms, the phrase can also be extended to gravitational observables that diagnose nuclear microphysics in neutron stars. One review explicitly develops a nuclear-to-gravitational chain in which laboratory constraints on the symmetry energy $V(z)=m_ngz$14 propagate into predicted moments of inertia, deformations, and continuous gravitational-wave amplitudes from pulsars (Krastev et al., 2010). A later inspiral study treats tidal deformabilities from binary-neutron-star mergers as a kind of “nuclear gravitational spectroscopy,” because the waveform carries information about dense-matter parameters such as $V(z)=m_ngz$15, though that mapping is limited by a systematic floor of about $V(z)=m_ngz$16 in the two-dimensional analysis and can be reduced to about $V(z)=m_ngz$17 using multidimensional correlations among tidal deformabilities at several masses (Carson et al., 2019). These works broaden the meaning from laboratory spectroscopy to gravitational inference of nuclear matter properties.

A further extension appears in neutron-capture $V(z)=m_ngz$18 spectroscopy for exotic invisible channels. A 2026 framework proposes searching for correlated “satellite-line combs” at a common offset $V(z)=m_ngz$19 below known capture transitions, a strategy that is explicitly said to be relevant to searches for graviton-like or gravity-coupled emission channels if such channels produce a discrete two-body precursor emission (Meirose et al., 27 Mar 2026). This suggests that, at its broadest, nuclear gravitational spectroscopy includes any spectroscopic architecture in which gravitational coupling or a gravity-like channel is inferred from correlated spectral structure in a nuclear or neutron system.

Taken together, the literature supports a layered definition. In its dominant current laboratory form, the subject is neutron gravitational spectroscopy: precision spectroscopy of gravitationally quantized external neutron motion above mirrors. In its stricter nuclear form, it is the gravitational spectroscopy of nuclear transitions, now reformulated as a heterodyne interferometric phase-estimation problem. In broader gravitational-physics usage, it also includes astrophysical and exotic-channel inference in which gravitational observables constrain nuclear structure or rare gravity-coupled processes.